How do I make the spacebar menu display nodes in the Node Editor?
I'm not sure if the node search by name is on default on the Spacebar. However you can assign it any shortcut you want.

By default, you can access the search by pressing &#8679; ShiftA > and then selecting Search. Your presented searchbar will show up and show the nodes as results.



(I consider it possible, that the makers of the tutorials you mention, simply pressed clicked the search option too fast for a frame to capture.)
If you wish to add an even more direct shortcut to this functionality, go to the User Preferences &#9096; Ctrl&#9095; AltU. Scroll down to the Node Editor and add a new shortcut. Assign a key combination to it and enter node.add_search into the identifier field.